and this is reflected in the fact that P ¼ 0. This is not commonly appreciated, but
follows from Newton’s second law: if the molecules are expanding freely into a
vacuum and do not change momentum through collisions with the wall of a
container they are not themselves subject to a force and cannot exert a force. We
therefore have an immediate conflict with the First Law that is built into the structure
of thermodynamics via Clausius’ assertion that, “The law does not speak of the work
which the heat does, but of the work which it can do . . .”. The idea that heat can do
work is a direct consequence of the view of the time that heat was somehow
converted into work, but we now know that work is a consequence of repeated
collisions of particles on a piston. Heat flows into a gas to replace the energy lost
during work and in this sense there is no such thing as isothermal heat flow.
However, borrowing from the terminology of heat engines prevalent at the time of
Clausius, we can regard heat as capable of doing work PδV. If, following Clausius,
we use this work term to give the increase in entropy during an irreversible
expansion, we are adding a term in energy that does not in fact reflect the physical
processes and violates energy conservation.
This failure of the mathematics to reflect the physics has been overlooked by the
majority of physicists for well over 160 years and consideration of this alone shows
that the process of interpreting mathematical formalisms in terms of physics is not
straightforward. It is perhaps not surprising that students struggle. It is over 10 years
since Rebello, working with Dean Zollmann and others (Rebello et al. 2005), looked
at the transfer of mathematical knowledge from one domain to another, but little
progress seems to have made since then. Authors such as Karam (2014) and Redish
and Gupta (2010) stress that understanding mathematics in physics is not just about
understanding mathematical operations, but how those operations connect to and
describe physics concepts. I suggest that we have only just begun to understand
some of the complex interactions in not just learning physics, but in doing it and that
this process of examining critically the way physics is done should impact on our
understanding of the fundamentals.
Revisiting fundamental concepts and their connection to mathematical formalisms means keeping an open mind and rejecting a utilitarian approach. That is, just
because a mathematical approach appears to be useful does not mean it is correct and
it has been argued in this chapter that Clausius’ conception of entropy was flawed in
so far as it was based around the concept of transformations rather than conservation
of energy. Clearly, the difficulties engendered by this approach were not realised at
the time and the failure to reflect on the disconnection between the mathematics and
the physics has left its mark on thermodynamics today. That leads inevitably to the
question of what might usefully be taught in thermodynamics.
It is my own personal view that a fundamental revision of the foundations of the
subject is required. Carnot was concerned with the reversibility of the cycle itself,
which was also the view of Kelvin: the cycle can be performed in one direction to
convert heat into work or in the other to convert work into heat. This view of
thermodynamics has been overlooked in favour of Clausius, but the implication of
the work summarised here is that this view is actually correct. There is a powerful
argument, therefore, for returning to the origins of the subject and basing
11 Physics Education Research and the Foundations of Physics: A Case Study from. . .
125
follows from Newton’s second law: if the molecules are expanding freely into a
vacuum and do not change momentum through collisions with the wall of a
container they are not themselves subject to a force and cannot exert a force. We
therefore have an immediate conflict with the First Law that is built into the structure
of thermodynamics via Clausius’ assertion that, “The law does not speak of the work
which the heat does, but of the work which it can do . . .”. The idea that heat can do
work is a direct consequence of the view of the time that heat was somehow
converted into work, but we now know that work is a consequence of repeated
collisions of particles on a piston. Heat flows into a gas to replace the energy lost
during work and in this sense there is no such thing as isothermal heat flow.
However, borrowing from the terminology of heat engines prevalent at the time of
Clausius, we can regard heat as capable of doing work PδV. If, following Clausius,
we use this work term to give the increase in entropy during an irreversible
expansion, we are adding a term in energy that does not in fact reflect the physical
processes and violates energy conservation.
This failure of the mathematics to reflect the physics has been overlooked by the
majority of physicists for well over 160 years and consideration of this alone shows
that the process of interpreting mathematical formalisms in terms of physics is not
straightforward. It is perhaps not surprising that students struggle. It is over 10 years
since Rebello, working with Dean Zollmann and others (Rebello et al. 2005), looked
at the transfer of mathematical knowledge from one domain to another, but little
progress seems to have made since then. Authors such as Karam (2014) and Redish
and Gupta (2010) stress that understanding mathematics in physics is not just about
understanding mathematical operations, but how those operations connect to and
describe physics concepts. I suggest that we have only just begun to understand
some of the complex interactions in not just learning physics, but in doing it and that
this process of examining critically the way physics is done should impact on our
understanding of the fundamentals.
Revisiting fundamental concepts and their connection to mathematical formalisms means keeping an open mind and rejecting a utilitarian approach. That is, just
because a mathematical approach appears to be useful does not mean it is correct and
it has been argued in this chapter that Clausius’ conception of entropy was flawed in
so far as it was based around the concept of transformations rather than conservation
of energy. Clearly, the difficulties engendered by this approach were not realised at
the time and the failure to reflect on the disconnection between the mathematics and
the physics has left its mark on thermodynamics today. That leads inevitably to the
question of what might usefully be taught in thermodynamics.
It is my own personal view that a fundamental revision of the foundations of the
subject is required. Carnot was concerned with the reversibility of the cycle itself,
which was also the view of Kelvin: the cycle can be performed in one direction to
convert heat into work or in the other to convert work into heat. This view of
thermodynamics has been overlooked in favour of Clausius, but the implication of
the work summarised here is that this view is actually correct. There is a powerful
argument, therefore, for returning to the origins of the subject and basing
11 Physics Education Research and the Foundations of Physics: A Case Study from. . .
125